scholarly journals A Review on the Qualitative Behavior of Solutions in Some Chemotaxis–Haptotaxis Models of Cancer Invasion

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1464
Author(s):  
Yifu Wang
Keyword(s):  
Differential Equations ◽  
Cancer Invasion ◽  
Point Of View ◽  
Qualitative Behavior ◽  
Embryonic Cells ◽  
Open Problems ◽  
Strongly Coupled ◽  
Invasion Models ◽  
Oriented Movement ◽  
Important Extension

Chemotaxis is an oriented movement of cells and organisms in response to chemical signals, and plays an important role in the life of many cells and microorganisms, such as the transport of embryonic cells to developing tissues and immune cells to infection sites. Since the pioneering works of Keller and Segel, there has been a great deal of literature on the qualitative analysis of chemotaxis systems. As an important extension of the Keller–Segel system, a variety of chemotaxis–haptotaxis models have been proposed in order to gain a comprehensive understanding of the invasion–metastasis cascade. From a mathematical point of view, the rigorous analysis thereof is a nontrivial issue due to the fact that partial differential equations (PDEs) for the quantities on the macroscale are strongly coupled with ordinary differential equations (ODEs) modeling the subcellular events. It is the goal of this paper to describe recent results of some chemotaxis–haptotaxis models, inter alia macro cancer invasion models proposed by Chaplain et al., and multiscale cancer invasion models by Stinner et al., and also to introduce some open problems.

scholarly journals On the qualitative behavior of the solutions to second-order neutral delay differential equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shyam Sundar Santra ◽  
Hammad Alotaibi ◽  
Omar Bazighifan
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Qualitative Behavior ◽  
Multiple Delays ◽  
Neutral Delay ◽  
Open Problems ◽  
Neutral Delay Differential Equations ◽  
Delay Differential ◽  
Necessary And Sufficient

AbstractDifferential equations of second order appear in numerous applications such as fluid dynamics, electromagnetism, quantum mechanics, neural networks and the field of time symmetric electrodynamics. The aim of this work is to establish necessary and sufficient conditions for the oscillation of the solutions to a second-order neutral differential equation. First, we have taken a single delay and later the results are generalized for multiple delays. Some examples are given and open problems are presented.

scholarly journals New Conditions for the Oscillation of Second-Order Differential Equations with Sublinear Neutral Terms

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1159
Author(s):  
Shyam Sundar Santra ◽  
Omar Bazighifan ◽  
Mihai Postolache
Keyword(s):  
Fluid Dynamics ◽  
Neural Networks ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Qualitative Behavior ◽  
Neutral Differential Equations ◽  
Second Order Differential Equations ◽  
Delay Differential

In continuous applications in electrodynamics, neural networks, quantum mechanics, electromagnetism, and the field of time symmetric, fluid dynamics, neutral differential equations appear when modeling many problems and phenomena. Therefore, it is interesting to study the qualitative behavior of solutions of such equations. In this study, we obtained some new sufficient conditions for oscillations to the solutions of a second-order delay differential equations with sub-linear neutral terms. The results obtained improve and complement the relevant results in the literature. Finally, we show an example to validate the main results, and an open problem is included.

scholarly journals New Results on Qualitative Behavior of Second Order Nonlinear Neutral Impulsive Differential Systems with Canonical and Non-Canonical Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 934
Author(s):  
Shyam Sundar Santra ◽  
Khaled Mohamed Khedher ◽  
Kamsing Nonlaopon ◽  
Hijaz Ahmad
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Discrete Equation ◽  
Qualitative Behavior ◽  
Differential Systems ◽  
Impulsive Differential Systems

The oscillation of impulsive differential equations plays an important role in many applications in physics, biology and engineering. The symmetry helps to deciding the right way to study oscillatory behavior of solutions of impulsive differential equations. In this work, several sufficient conditions are established for oscillatory or asymptotic behavior of second-order neutral impulsive differential systems for various ranges of the bounded neutral coefficient under the canonical and non-canonical conditions. Here, one can see that if the differential equations is oscillatory (or converges to zero asymptotically), then the discrete equation of similar type do not disturb the oscillatory or asymptotic behavior of the impulsive system, when impulse satisfies the discrete equation. Further, some illustrative examples showing applicability of the new results are included.

Deterministic Stability Analysis of a Wind Loaded Structure

1978 ◽  
Vol 45 (1) ◽  
pp. 165-169 ◽  
Author(s):  
P. J. Holmes ◽  
Y. K. Lin
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Wind Loading ◽  
Qualitative Behavior ◽  
Stability Boundaries ◽  
Quantitative Estimates ◽  
Loading Problem ◽  
Loaded Structure ◽  
Excitation Conditions

We discuss the qualitative behavior of a pair of nonlinear differential equations arising in the study of a wind loading problem when turbulence terms are ignored. We obtain quantitative estimates of stability boundaries and are able to identify the most dangerous excitation conditions. This deterministic study provides the basis for further work on the full stochastic differential equations resulting when turbulence terms are included.

scholarly journals The combinatorics of discrete time-trees: theory and open problems

2016 ◽  
Author(s):  
Alex Gavryushkin ◽  
Chris Whidden ◽  
Frederick A Matsen
Keyword(s):  
Shortest Paths ◽  
Point Of View ◽  
Upper And Lower Bounds ◽  
Evolutionary Divergence ◽  
Discrete Approximations ◽  
Open Problems ◽  
Graph Theoretic ◽  
Tree Graphs ◽  
Object Of Study ◽  
Rooted Phylogenetic Tree

ABSTRACTA time-tree is a rooted phylogenetic tree such that all internal nodes are equipped with absolute divergence dates and all leaf nodes are equipped with sampling dates. Such time-trees have become a central object of study in phylogenetics but little is known about the parameter space of such objects. Here we introduce and study a hierarchy of discrete approximations of the space of time-trees from the graph-theoretic and algorithmic point of view. One of the basic and widely used phylogenetic graphs, the NNI graph, is the roughest approximation and bottom level of our hierarchy. More refined approximations discretize the relative timing of evolutionary divergence and sampling dates. We study basic graph-theoretic questions for these graphs, including the size of neighborhoods, diameter upper and lower bounds, and the problem of finding shortest paths. We settle many of these questions by extending the concept of graph grammars introduced by Sleator, Tarjan, and Thurston to our graphs. Although time values greatly increase the number of possible trees, we show that 1-neighborhood sizes remain linear, allowing for efficient local exploration and construction of these graphs. We also obtain upper bounds on the r-neighborhood sizes of these graphs, including a smaller bound than was previously known for NNI.Our results open up a number of possible directions for theoretical investigation of graph-theoretic and algorithmic properties of the time-tree graphs. We discuss the directions that are most valuable for phylogenetic applications and give a list of prominent open problems for those applications. In particular, we conjecture that the split theorem applies to shortest paths in time-tree graphs, a property not shared in the general NNI case.

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